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Properties

Label 2-637-91.81-c1-0-39
Degree $2$
Conductor $637$
Sign $-0.898 + 0.438i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 − 2.01i)2-s + 2.30·3-s + (−1.71 − 2.97i)4-s + (−1.68 − 2.91i)5-s + (2.69 − 4.66i)6-s − 3.34·8-s + 2.33·9-s − 7.85·10-s + 2.33·11-s + (−3.96 − 6.87i)12-s + (0.408 + 3.58i)13-s + (−3.89 − 6.74i)15-s + (−0.466 + 0.808i)16-s + (2.72 + 4.72i)17-s + (2.71 − 4.70i)18-s − 7.16·19-s + ⋯
L(s)  = 1  + (0.824 − 1.42i)2-s + 1.33·3-s + (−0.858 − 1.48i)4-s + (−0.753 − 1.30i)5-s + (1.09 − 1.90i)6-s − 1.18·8-s + 0.777·9-s − 2.48·10-s + 0.702·11-s + (−1.14 − 1.98i)12-s + (0.113 + 0.993i)13-s + (−1.00 − 1.74i)15-s + (−0.116 + 0.202i)16-s + (0.661 + 1.14i)17-s + (0.640 − 1.10i)18-s − 1.64·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.898 + 0.438i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.898 + 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676485 - 2.92709i\)
\(L(\frac12)\) \(\approx\) \(0.676485 - 2.92709i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-0.408 - 3.58i)T \)
good2 \( 1 + (-1.16 + 2.01i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 + (1.68 + 2.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.33T + 11T^{2} \)
17 \( 1 + (-2.72 - 4.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 7.16T + 19T^{2} \)
23 \( 1 + (-3.22 + 5.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.22 - 7.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.52 + 2.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.52 - 2.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.468 - 0.812i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.04 + 3.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 2.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.17 + 2.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.62 + 6.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.39T + 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 + (-3.79 + 6.57i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.03 + 1.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.79 - 6.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.89T + 83T^{2} \)
89 \( 1 + (6.57 - 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.77 + 3.08i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.34646915106989752045656241382, −9.224696809189143224631623436320, −8.707407580995690726626819094821, −8.071469579731047115498061627822, −6.52837834644259138651705455720, −4.93278341097878210986699067128, −4.13669096871817833515954357938, −3.64912635468989981530883328805, −2.31397630469855369914719347403, −1.27037103182144382545279948801, 2.73676171278023224283348467809, 3.49634030185441026800058464471, 4.29659977058166801584180337669, 5.70098372095127071276472394142, 6.73093557353715359927386363843, 7.38035334435567519582866088482, 8.005633854625212988399601246203, 8.726902099445535695450253240832, 9.880552149897001796674441701050, 10.95939121275141527761186613479

Graph of the $Z$-function along the critical line